160 XIII INTEGRATION the set of all real-valued ^-measurable functions / such that/2 is integrable is a vector space g&X, u) (also denoted by «£?£(//) or && For if / and g are measurable and/2 and g2 are integrable, then/+# is measurable by virtue of (13.11.2.1), and it follows from (13.9.13) that (/ + g)2 is integrable. There exist functions/which are non-measurable, such that/2 is integrable (Problem 2). By abuse of language, the space S£ R(X, u) is called the space of square- integrable functions. Since Np(af) = \a\Np(f) for p=l,2 and any scalar a 7* 0, it follows from (13.11.2.1) that Np is a seminorm on the space J$?£(X, u). The set Jf of functions / such that N//) = 0 is in both cases the vector subspace of negligible (finite) real- valued functions (13.6.3). Hence the space JS?£(X, jj) is not in general Hausdorff with respect to the topology defined by the seminorm Np . The quotient space L£(X, ju) = J?fi(X, ju)/N (also denoted by LJGu) or Lg) is the space of equivalence classes / of integrable functions when p = 1, square-integrable functions when /? = 2 (13.6). A function which is defined and finite almost everywhere in X is said to be square-integrable if its class belongs to LR . The number N//) is the same for all functions / belonging to a class /eL£, and is denoted by Np(/). From (12.14.8) it follows that /H-»NP(/) is a norm on L£ (p — 1 , 2). (13.11.4) (Fischer-Riesz Theorem) The normed space LJ(X, u) is com- plete (p = 1, 2) (in other words, it is a Banach space). More precisely: (i) .//* (gn) is a sequence of functions on X whose classes belong to Lg, oo < + oo, £/ze« ?//e jenas whose general term is gn(x) converges absolutely in R almost everywhere. If g(x) = £ gn(x), the class g of the func- n~ 1 oo tion g (defined almost everywhere) belongs to Lg , and we have g = £ gn n= 1 in the normed space LJ . (ii) If (/„) is a sequence of functions such that the sequence (/„) of their equivalence classes is a Cauchy sequence in L£ , then there exists a subsequence (/nj such that (fnk(x)) converges to a limit f(x) in R almost everywhere. For each such subsequence (fn]), the class ? of f belongs to L£ and is the limit of the sequence (/„) in the normed space Lg . (iii) Let (/„) be a sequence of functions belonging to JS?g(X, u) such that the sequence (fn(x)) converges almost everywhere to a limit /(x), and suppose that there exists a function h e ^g(X, u) such that \fn(x)\ g h(x) almost every- where, for all n. Then the class / off belongs to L£ and is the limit of the sequence (/„) in the normed space L£ .Use (a) to prove that the condition is necessary.)